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We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach numbers Mt and Taylor-scale Reynolds numbers Reλ. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as that in incompressible flow due to density-velocity coupling. The modulus of the smallest Lyapunov exponent λ3 provides the principal Lyapunov exponent of the time-reversed flow, which is usually wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent λ1, determines all the exponents due to vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier–Stokes equations and tracking particles verify these theoretical predictions. We found that (1) the largest normalized Lyapunov exponent λ1τη, where τη is the Kolmogorov timescale, is a decreasing function of Mt. Its dependence on Reλ is weak when the driving force is solenoidal, while it is an increasing function of Reλ when the solenoidal and compressible forces are comparable. Similar facts hold for |λ3|, in contrast to well-studied short-correlated model; (2) the ratio of the first two Lyapunov exponents λ1/λ2 decreases with Reλ and is virtually independent of Mt for Mt≤1 in the case of solenoidal force but decreases as Mt increases when solenoidal and compressible forces are comparable; (3) for purely solenoidal force, λ1:λ2:λ3≈4:1:−5 for Reλ>80, which is consistent with incompressible turbulence studies; and (4) the ratio of dilation-to-vorticity is a more suitable parameter to characterize Lyapunov exponents than Mt.
We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach numbers Mt and Taylor-scale Reynolds numbers Reλ. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as that in incompressible flow due to density-velocity coupling. The modulus of the smallest Lyapunov exponent λ3 provides the principal Lyapunov exponent of the time-reversed flow, which is usually wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent λ1, determines all the exponents due to vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier–Stokes equations and tracking particles verify these theoretical predictions. We found that (1) the largest normalized Lyapunov exponent λ1τη, where τη is the Kolmogorov timescale, is a decreasing function of Mt. Its dependence on Reλ is weak when the driving force is solenoidal, while it is an increasing function of Reλ when the solenoidal and compressible forces are comparable. Similar facts hold for |λ3|, in contrast to well-studied short-correlated model; (2) the ratio of the first two Lyapunov exponents λ1/λ2 decreases with Reλ and is virtually independent of Mt for Mt≤1 in the case of solenoidal force but decreases as Mt increases when solenoidal and compressible forces are comparable; (3) for purely solenoidal force, λ1:λ2:λ3≈4:1:−5 for Reλ>80, which is consistent with incompressible turbulence studies; and (4) the ratio of dilation-to-vorticity is a more suitable parameter to characterize Lyapunov exponents than Mt.
Turbulence in the atmospheric surface layer, especially in deserts and semi-arid regions, significantly affects sand movement. In unstable stratification, turbulence exhibits complex intermittency, complicating its impact on saltation. This study uses wavelet transform analysis to examine the effects of turbulence intermittency in unstable stratification on saltation. Our analysis reveals that in unstable stratification, the energy distribution of turbulence is more dispersed, the intermittent characteristics are more significant, and the intermittent burst duration of streamwise turbulence is longer, while the vertical intermittent burst duration is shorter. The fitting formulas of the energy ratio and stratification stability of the streamwise wind speed, vertical wind speed, and temperature at different frequencies are given. In addition, there is a complex nonlinear relationship between stratification stability and friction velocity on saltation. In unstable stratification, the critical wind speed required for saltation is higher than that of near-neutral, and the jumping speed and horizontal transport are weakened. Moreover, the coherence between wind speed and saltation flux increases significantly at low frequency with the increase in instability, indicating that large-scale motion plays a key role in saltation under these conditions. The more unstable the stratification is, the more obvious the phase difference fluctuation of the low frequency part is, and the more unfavorable the formation of stable saltation sand conditions. This study reveals the turbulence intermittently and its complex effects on sand particle movement in unstable stratification, which is of great significance for predicting and controlling dust storms, land desertification, and soil erosion.
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